SSAT A new characterization of NP

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SSAT A new characterization of NP

• and the hardness of approximating CVP.

joint work with G. Kindler, R. Raz, and S. Safra
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Lattice Problems

• Definition Given v1,..,vk?Rn,
• The lattice LL(v1,..,vk) ?aivi integers
  ai
• SVP Find the shortest non-zero vector in L.
• CVP Given a vector y?Rn, find a v?L closest to
  y.

y
shortest
closest
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Lattice Approximation Problems

• g-Approximation version Find a vector whose
  distance is at most g times the optimal distance.
• g-Gap version Given a lattice L, a vector y,
  and a number d, distinguish between
• The yes instances (dist(y,L)ltd)
• The no instances (dist(y,L)gtgd)
• If g-Gap problem is NP-hard, then having a
  g-approximation polynomial algorithm --gt PNP.

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Lattice Problems - Brief History

• Dirichlet, Minkowsky no CVP algorithms
• LLL Approximation algorithm for SVP, factor
  2n/2
• Babai Extension to CVP
• Schnorr Improved factor, (1?)n for both CVP
  and SVP
• vEB CVP is NP-hard
• ABSS Approximating CVP is
• NP hard to within any constant
• Quasi NP hard to within an almost polynomial
  factor.

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Lattice Problems - Recent History

• Ajtai96 average-case/worst-case equiv. for
  SVP.
• Ajtai-Dwork96 Cryptosystem
• Ajtai97 SVP is NP-hard (for randomized
  reductions).
• Micc98 SVP is NP-hard to approximate to within
  some constant factor.
• LLS Approximating CVP to within n1.5 is in
  coNP.
• GG Approximating SVP and CVP to within ?n is
  in coAM?NP.

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Lattice Problems

• Definition Given v1,..,vk?Rn,
• The lattice LL(v1,..,vk) ?aivi integers
  ai
• SVP Find the shortest non-zero vector in L.
• CVP Given a vector y?Rn, find a v?L closest to
  y.

y
shortest
closest
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Reducing g-SVP to g-CVP GMSS98
The lattice L
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Reducing g-SVP to g-CVP GMSS98
CVP oracle apx. minimize c1b12c2b2-b2
Lspan (2b1,b2)
Lspan (b1,2b2)
Note at least one coef. ci of the shortest
vector must be odd
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The Reduction
Input A pair (B,d), B(b1,..,bn) and d?R
for j1 to n invoke the CVP oracle
on(B(j),bj,d) Output The OR of all oracle
replies.
Where B(j) (b1,..,bj-1,2bj,bj1,..,bn)
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SSATA new Characterization of NP

• and the hardness of approximating CVP

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Hardness of approx. CVP DKRS

• g-CVP is NP-hard for gn1/loglog n
• n - lattice dimension
• Improving
• Hardness (NP-hardness instead of
  quasi-NP-hardness)
• Non-approximation factor (from 2(logn)1-?)

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• ABSS reduction uses PCP to show
• NP-hard for gO(1)
• Quasi-NP-hard g2(logn)1-? by repeated blow-up.
• Barrier - 2(logn)1-? ?const ?gt0
• SSAT a new non-PCP characterization of NP.
• NP-hard to approximate to within gn1/loglogn .

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SAT

• Input ?f1,..,fn Boolean functions tests
• x1,..,xn variables with range 0,1
• Problem Is ? satisfiable?
• Thm (Cook-Levin) SAT is NP-complete
• (even when depend(?)3)

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SAT as a consistency problem

• Input
• ?f1,..,fn Boolean functions - tests
• x1,..,xn variables with range R
• for each test a list of satisfying assignments
• Problem
• Is there an assignment to the tests that is
  consistent?

f(x,y,z)
g(w,x,z)
h(y,w,x)
(1,0,7) (1,3,1) (3,2,2)
(0,2,7) (2,3,7) (3,1,1)
(0,1,0) (2,1,0) (2,1,5)
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Super-Assignments
A natural assignment for f(x,y,z)
A(f) (3,1,1)
1 0
(1,1,2) (3,1,1) (3,2,5) (3,3,1) (5,1,2)
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Consistency
In the SAT case
A(f) (3,2,5) A(f)x (3)
?x ? f,g that depend on x A(f)x A(g)x
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Consistency
SA(f) 3(1,1,2) ? -2(3,2,5) ? 2(3,3,1)
Consistency ?x ? f,g that depend on x SA(f)x
SA(g)x
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g-SSAT - Definition

• Input
• ?f1,..,fn tests over variables x1,..,xn with
  range R
• for each test fi - a list of sat. assign.
• Problem Distinguish between
• Yes There is a natural assignment for ?
• No Any non-trivial consistent super-assignment
  is of norm gt g
• Theorem SSAT is NP-hard for gn1/loglog n.
• (conjecture gn? , ? some constant)

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Attempt at reducing PCP to SSAT

• Take a PCP test-system ? f1,...,fn

No instances
Yes instances
? Assignment (to vars.) satisfies only ?
fraction of ?
the GAP
? Satisfying assignment for ?
Is there a super-assignment for a no
instance, consistent small-norm (less than
gn1/loglog n)
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A PCP no-instance
g(x,z)
h(y,z)
f(x,y)
(1,2) (2,2) (2,1)
(1,3) (3,3) (3,1)
(1,5) (5,5) (5,1)
Best assignment satisfies 2/3 of ? f,g,h x
lt--- 1 y lt--- 2 z lt--- 3
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An SSAT almost-yes-instance
g(x,z)
h(y,z)
f(x,y)
(1,2) (2,2) (2,1)
(1,3) (3,3) (3,1)
(1,5) (5,5) (5,1)
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f( x0 x1 )
1 (1 2)
-1 (2 2)
1 (2 1)
1(1)
1(1)
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f( x0 x1 x2 x3 x4 x5 x6 )
1 (1 2 3 4 5 6 0 )
-1 (2 2 2 2 2 2 2 )
1 (2 1 0 6 5 4 3 )
1(1)
1(1)
mod 7
linear extension
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Low Degree Extension

• embed variables in a domain 1..hd
• extend the domain 1..pd (p?h3, prime)

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Consistently Reading an LDF

• Replace each test with several new tests
  depending on the original variables and some new
  extension variables.
• satisfying assignment a Low-Degree-Extension

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Suppose we had...

• Consistency Lemma
• low-norm super-assignment for tests
• --gt ? global super-LDF
• that agrees with the tests.
• Deduce a satisfying assignment for almost
  all of ?s tests.

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A Consistent-Reader for LDFs
using composition-recursion

• Short representation.
• Negligible error.

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Representing a degree-h LDF

• in one piece, by writing its coefficients
• there are too many degree-h polynomials
• there are ? ph such polynomials
• (where h n1/loglogn, p ? h3).
• in many smaller pieces

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A Consistency Lemma
cube constant-dimensional affine subspace
test
test
test
test
test
test
Consistency For every pair of cubes with mutual
points -- their super-LDFs agree.
?Global super-LDF Agreeing with the cubes
super-LDFs
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Embedding Extension
X1 X2 X3
x
f(.)x5y2
y1 y2 y3
y
(x, x2, x4, y, y2, y4)
(x,y)
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A Tree of Consistent Readers
The low-degree-extension domain
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SSAT is NP-hard to approximateto within g
n1/loglogn
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Reducing SSAT to CVP
f,(1,2)
f,(3,2)
w w w w w w w w
1 2 3
f,f,x
0 0 0 0 0 0 0 0
f(w,x) f(z,x)
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A consistency gadget
0 0 w 0
w w w w
w w 0 w
1 2 3
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A consistency gadget
0 0 w 0
w w w w
w w 0 w
w 0 w w
0 0 0 w
0 w 0 0
w w w 0
1 2 3
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Conclusion

• SSAT is NP-hard to approx. to within
• gn1/loglog n
• CVP is NP-hard to approximate to within
• the same g
• Future Work
• Increase to gnc, c constant.
• Extend CVP to SVP reduction